A dual hybrid virtual element method for plane elasticity problems

Abstract

A dual hybrid Virtual Element scheme for plane linear elastic problems is presented and analysed. In particular, stability and convergence results have been established. The method, which is first order convergent, has been numerically tested on two benchmarks with closed form solution, and on a typical microelectromechanical system. The numerical outcomes have proved that the dual hybrid scheme represents a valid alternative to the more classical low-order displacementbased Virtual Element Method.

Stress‐hybrid virtual element method on quadrilateral meshes for compressible and nearly‐incompressible linear elasticity

“…Initially, mixed variational principles, hybrid formulations, B-bar and selective reduced integration strategies that are prominent in finite element formulations for constrained problems have been adopted in the virtual element method. [4][5][6][7][8] Böhm et al 9 has provided a study of different virtual element methods for incompressible problems and compared the results to classical finite element techniques. More recently, in the spirit of assumed-strain methods, 10,11 projections onto higher order strains have been pursued in the VEM to devise stabilization-free schemes.…”

“…The advent of the virtual element method 2,3 has provided new routes to potentially alleviate locking for nearly‐incompressible materials. Initially, mixed variational principles, hybrid formulations, B‐bar and selective reduced integration strategies that are prominent in finite element formulations for constrained problems have been adopted in the virtual element method 4‐8 . Böhm et al 9 has provided a study of different virtual element methods for incompressible problems and compared the results to classical finite element techniques.…”

“…The recent development of the Virtual Element Method (VEM) [6,7,8] has provided an alternative way to extend these methods to alleviate volumetric locking for general polygons and polyhedra. Methods such as mixed and hybrid formulations [3,5,26,28], B-bar [49], assumed strain [11,18,19,29,41], and recently hybrid stress for general quadrilaterals [20] have been adopted in the virtual element method. The approach of using triangular subelements has also been applied in [71] to construct a stabilization term for the virtual element method.…”

Design and analysis of the Extended Hybrid High-Order method for the Poisson problem

“…However, the analysis requires a discrete trace inequality [2,Equation (49)], which depends on the singular function. Following this approach, the same authors have defined an enriched NC-VEM for a plane elasticity problem with corner singularities [1]. An extended hybrid high-order method for the Poisson problem which avoids the use of discrete inequalities dependent on the enrichment function is provided by Yemm [21].…”

A virtual element method for the elasticity problem allowing small edges

“…The bibliography related to numerical methods to approximate the displacement of some elastic structure is abundant, where different method as been proposed, as [5,11,13,15,18,19,24], where mixed finite element methods, mixed virtual element methods, stabilized methods, discontinuous Galerkin methods, just for mention some of them, have been considered. Regarding the study of VEM applied to elasticity problems, we can cite the following works [3,4,20,21,22,25,26]. In particular, a VEM for the primal formulation of the elasticity eigenvalue problem as been proposed in [20], where the proposed virtual spaces on this reference are a suitable alternative for the load problem.…”